FIG. 1 represents a simplified target tracking system 10. System 10 tracks a target, illustrated as being an aircraft 12, by the use of multiple radar systems 14, 16. Radar system 14 includes a radar antenna 14a, which transmits and receives radar signals illustrated by “lightning bolt” symbols 18. Portions of the transmitted signals 18 are reflected by target 12 and return to the radar antenna 14a. The returned signals allow the generation of measurements at an output port 14o of radar system 14. Radar system 16 includes a radar antenna 16a, which transmits and receives radar signals illustrated by “lightning bolt” symbols 20. Portions of the transmitted signals 20 are reflected by target 12 and return to the radar antenna 16a. The returned signals allow the generation of measurements at an output port 16o of radar system 16. These measurements include values of at least target position, possibly in the form of range and angles from the radar systems 14 and 16. A possible scenario is that radar systems 14 and 16 have measurements which are corrupted by unknown random measurement noises, characterized by a covariance and unknown time-varying biases with known bounds. The biases may be the result of sensor misalignment. The measurements are applied to a processing arrangement 22, which determines from the measurements various target parameters, which may include course (direction of motion), speed, and target type. The estimated position of the target, and possibly other information, is provided to a utilization apparatus or user, illustrated in this case as being a radar display 24. The operator (or possibly automated decision-making equipment) can make decisions as to actions to be taken in response to the displayed information. It should be understood that the radar tracking system 10 of FIG. 1 is only one embodiment of a general class of estimation systems for systems with distributed sensors such as nuclear, chemical, or manufacturing factories or facilities, control processes subject to external parameter changes, space station subject to vibrations, automobile subject to weather conditions, and the like.
State-of-the-art tracking systems utilize measurements fed to a processing site from multiple sensors. These sensors may have different measuring accuracies (i.e., random errors) and unknown measurement biases that may be time-varying within physical bounds.
Consider the problem of tracking an airplane whose trajectory in three dimensions is an arbitrary curve with bounded instantaneous turn rate and tangential acceleration. The parameters of this tracking problem are the turn rate ω (which can be related to the curvature of the trajectory) and the tangential acceleration α. Sensors, such as multiple radars, observe the position of this airplane. Each sensor is subject to alignment errors which cause it to be rotated by an unknown amount from its nominal alignment, that is a consequence of imperfect mechanical mounting, flexure of the array structure due to temperature effects etc. These small or infinitesimal rotations constitute a vector b. The parameters, ω, α, and b are neither exclusively constant nor strictly white noise stochastic processes, but vary arbitrarily in time within physical bounds.
This problem belongs to a more general problem of estimating the state of a system using biased measurements. In prior art, the Kalman filter solves this problem in some situations by including the biases as part of an augmented state to be estimated. Such a filter is termed herein a “full state” estimator. An example of the Kalman filter bias estimation approach has been described in Y. Kosuge and T. Okada, “Minimum Eigenvalue Analysis Using Observation Matrix for Bias Estimation of Two 3- Dimensional Radars,” Proceedings of the 35th SICE Annual Conference, pp. 1083-1088, July 1996, Y. Kosuge and T. Okada, “Bias Estimation of Two 3-Dimensional Radars Using Kalman Filter,” Proceedings of the 4th International Workshop on Advanced Motion Control, pp. 377-382, March 1996, N. Nabaa and R. H. Bishop, “Solution to a Multisensor Tracking Problem with Sensor Registration Errors,” IEEE Transactions on Aerospace and Electronic Systems, pp. 354-363, Vol. 35, No. 1, January 1999, and E. J. Dela Cruz, A. T. Alouani, T. R. Rice, and W. D. Blair, “Estimation of Sensor Bias in Multisensor Systems,” Proceedings of IEEE Southeastcon 1992, pp. 210-214, Vol. 1, Apr. 12-15, 1992. However, the biases may vary too erratcially to be considered as observables. In the case in which biases cannot be estimated, filters, which do not augment the state vector with these parameters, often give better performance. Such a filter is termed herein a “reduced state” estimator. More generally, a “reduced state” or “reduced order” estimator uses fewer states than would be required to completely specify the system.
Difficulties of using a Kalman filter in other contexts, dealing with unknown time-varying bounded parameters affecting system dynamics are discussed in copending patent applications entitled “REDUCED STATE ESTIMATOR FOR SYSTEMS WITH PHYSICALLY BOUNDED PARAMETERS” and “REDUCED STATE ESTIMATION WITH MULTISENSOR FUSION AND OUT-OF-SEQUENCE MEASUREMENTS”, filed on or about Mar. 16, 2005 and Mar. 30, 2005, respectively, both in the names of P. Mookerjee and F. Reifler. These difficulties are also found in P. Mookerjee and F. Reifler, “Reduced State Estimators for Consistent Tracking of Maneuvering Targets,” IEEE Transactions on Aerospace and Electronic Systems (in press). P. Mookerjee and F. Reifler in “Reduced State Estimator for Systems with Parametric Inputs,” IEEE Transactions on Aerospace and Electronic Systems, pp. 446-461, Vol. AE-40, No. 2, April 2004.
The method of the prior art is to estimate the sensor bias by way of a Kalman filter using state augmentation. Improved or alternative estimators are desired for coping with biased measurements.